Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator |
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Authors: | M.V. Berry |
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Affiliation: | Department of Physics, Technion, Haifa, Israel |
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Abstract: | The Hamiltonian H = 2πp + V(θ) Σ∞−∞ δ(t − n)(0 θ < 2π) is solved exactly, classically and quantally; the so lutions depend strongly on . There is no classical chaos and the phase cylinder p, θ is filled with invariant curves, which are finite loops around the cylinder if is sufficiently irrational and are translates of the infinitely long p axis if is rational. Quantal quasi-energy states correspond exactly to these invariant curves: localized in p and extended in θ if is sufficiently irrational, and extended in p and localized in θ if is rational. For a classical or quantal initial pure-momentum state, the energy at time t = n grows as n2 if is rational (resonance) and remains bounded if is sufficiently irrational (non-resonance). If is very nearly rational (marginal resonance), the energy may grow as nλ where λ is expressed in terms of exponents describing the irrationality of and the continuity class of V(θ). If the value of is uncertain, ensemble-averaging over shows that the energy grows ultimately as n, i.e. diffusively, as though under random impulses. |
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